Oscillation Criteria for Certain Even Order Neutral Delay Differential Equations with Mixed Nonlinearities

Abstract

We establish some oscillation criteria for the following certain even order neutral delay differential equations with mixed nonlinearities: ${\left(r\left(t\right){\left|z^{\left(n-1\right)}\left(t\right)\right|}^{\alpha -1}{z}^{\left(n-1\right)}\left(t\right)\right)}^{\text{'}}+q_0(t){\left|(x\left({\tau }_0\left(t\right)\right)\right|}^{\alpha -1}x({\tau }_0(t))+q_1\left(t\right){\left|(x({\tau }_1(t))\right|}^{\beta -1}x({\tau }_1(t))+q_2\left(t\right){\left|(x({\tau }_2(t))\right|}^{\gamma -1}x({\tau }_2(t))=0,\mathrm{ }t\ge t_0,$ where $z(t)=x(t)+p(t)x(\sigma (t)),$ $n$ is even integer, and $\gamma >\alpha >\beta >0.$ Our results generalize and improve some known results for oscillation of certain even order neutral delay differential equations with mixed nonlinearities.